Scientists and technicians of virtually all disciplinary backgrounds were sought for the Manhattan Project. Physicists, chemists, metallurgists — these were in high demand at the production and research sites, but there are cameos by any kind of scientist you can imagine. Biologists? Check — need them for study of the effects of radiation on living organisms. Meteorologists? Check — need them to help forecast weather conditions before using or testing the bombs. Mathematicians? Check — useful for certain types of very difficult calculations.

I'm trying to think of any branch of science that was not represented somewhere. I am struggling. I suppose if one dug down into the sub-specializations there are people who would be less useful, but really, they would take anyone with even a modicum of ability of find some use for them, even if it was doing something very tedious (like just running through very boring equations or experiments).

They were in competition with other wartime projects, notably the MIT Radiation Laboratory but may others beside that, for personnel, so they were always fighting for more, always trying to keep the ones they had, always trying to keep the less-important ones from being drafted. It was a very "greedy" organization when it came to technical personnel.

As for what kinds of characteristics, they often ended up with young scientists, mostly men (but not always). Usually they had to be American citizens but there were some exceptions. They had to pass a security background check but it was not as strict as the ones that would come in the Cold War. The National Academy of Sciences compiled a massive Roster of Scientific Personnel that could be used as something of a "catalog" for different types of scientific specialities. "Ah, we could use a few more spectrometrists," they might say, and could get a listing of everyone in the country who listed that as a specialty.

There are precious few cases of scientists who were considered but not taken on (Einstein is the most famous of these), or scientists who were at one point on it but were dropped for some reason (a few of Oppenheimer's students at Berkeley were deemed too much of a security risk and were quietly cut from the work). The general trends was in favor of more, more, more.